Higher Roots

Simplify Expressions with Higher Roots

Up to now, in this chapter we have worked with squares and square roots. We will now extend our work to include higher powers and higher roots.

Let’s review some vocabulary first.

\begin{matrix} We write: & We say: \\ n^{2} & n squared \\ n^{3} & n cubed \\ n^{4} & n to the fourth \\ n^{5} & n to the fifth \end{matrix}

The terms ‘squared’ and ‘cubed’ come from the formulas for area of a square and volume of a cube.

It will be helpful to have a table of the powers of the integers from $−5 to 5$ . See Figure 1.

This figure consists of two tables. The first table shows the results of raising the numbers 1, 2, 3, 4, 5, x, and x squared to the second, third, fourth, and fifth powers. The second table shows the results of raising the numbers negative one through negative five to the second, third, fourth, and fifth powers. The table first has five columns and nine rows. The second has five columns and seven rows. The columns in both tables are labeled, “Number,” “Square,” “Cube,” “Fourth power,” “Fifth power,” nothing, “Number,” “Square,” “Cube,” “Fourth power,” and “Fifth power.” In both tables, the next row reads: n, n squared, n cubed, n to the fourth power, n to the fifth power, nothing, n, n squared, n cubed, n to the fourth power, and n to the fifth power. In the first table, 1 squared, 1 cubed, 1 to the fourth power, and 1 to the fifth power are all shown to be 1. In the next row, 2 squared is 4, 2 cubed is 8, 2 to the fourth power is 16, and 2 to the fifth power is 32. In the next row, 3 squared is 9, 3 cubed is 27, 3 to the fourth power is 81, and 3 to the fifth power is 243. In the next row, 4 squared is 16, 4 cubed is 64, 4 to the fourth power is 246, and 4 to the fifth power is 1024. In the next row, 5 squared is 25, 5 cubed is 125, 5 to the fourth power is 625, and 5 to the fifth power is 3125. In the next row, x squared, x cubed, x to the fourth power, and x to the fifth power are listed. In the next row, x squared squared is x to the fourth power, x cubed squared is x to the fifth power, x squared to the fourth power is x to the eighth power, and x squared to the fifth power is x to the tenth power. In the second table, negative 1 squared is 1, negative 1 cubed is negative 1, negative 1 to the fourth power is 1, and negative 1 to the fifth power is negative 1. In the next row, negative 2 squared is 4, negative 2 cubed is negative 8, negative 2 to the fourth power is 16, and negative 2 to the fifth power is negative 32. In the next row, negative 4 squared is 16, negative 4 cubed is negative 64, negative 4 to the fourth power is 256, and negative 4 to the fifth power is negative 1024. In the next row, negative 5 squared is 25, negative 5 cubed is negative 125, negative 5 to the fourth power is 625, and negative 5 to the fifth power is negative 3125. — First through fifth powers of integers from $−5$ to $5 .$

Notice the signs in Figure 1. All powers of positive numbers are positive, of course. But when we have a negative number, the even powers are positive and the odd powers are negative. We’ll copy the row with the powers of $−2$ below to help you see this.

This figure has five columns and two rows. The first row labels each column: n, n squared, n cubed, n to the fourth power, and n to the fifth power. The second row reads: negative 2, 4, negative 8, 16, and negative 32.

Earlier in this chapter we defined the square root of a number.

If n^{2} = m, then n is a square root of m .

And we have used the notation $\sqrt{m}$ to denote the principal square root. So $\sqrt{m} \geq 0$ always.

We will now extend the definition to higher roots.

We do not write the index for a square root. Just like we use the word ‘cubed’ for $b^{3}$ , we use the term ‘cube root’ for $\sqrt[3]{a}$ .

We refer to Figure 1 to help us find higher roots.

\begin{matrix} 4^{3} & = & 64 & \sqrt[3]{64} & = & 4 \\ 3^{4} & = & 81 & \sqrt[4]{81} & = & 3 \\ {(−2)}^{5} & = & −32 & \sqrt[5]{−32} & = & −2 \end{matrix}

Could we have an even root of a negative number? No. We know that the square root of a negative number is not a real number. The same is true for any even root. Even roots of negative numbers are not real numbers. Odd roots of negative numbers are real numbers.

Simplify: ⓐ $\sqrt[3]{8}$ ⓑ $\sqrt[4]{81}$ ⓒ $\sqrt[5]{32}$ .

Solution

ⓐ

This table illustrates finding the cube root of 8, demonstrating its value and the mathematical reason.
	$\sqrt[3]{8}$
Since ${(2)}^{3} = 8 .$	$2$

ⓑ

This table presents the fourth root of 81, its resulting value, and the mathematical explanation for the solution.
	$\sqrt[4]{81}$
Since ${(3)}^{4} = 81 .$	$3$

ⓒ

Example showing the evaluation of the fifth root of 32, with its mathematical justification.
	$\sqrt[5]{32}$
Since ${(2)}^{5} = 32 .$	$2$

Simplify: ⓐ $\sqrt[3]{−64}$ ⓑ $\sqrt[4]{−16}$ ⓒ $\sqrt[5]{−243}$ .

Solution

ⓐ

Demonstrates the calculation of the cube root of -64 with a step-by-step explanation.
	$\sqrt[3]{−64}$
Since ${(−4)}^{3} = −64 .$	$−4$

ⓑ

This table demonstrates why the fourth root of -16 is not a real number, providing the mathematical expression, the reasoning, and the conclusion.
	$\sqrt[4]{−16}$
Think, ${(?)}^{4} = −16 .$ No real number raised to the fourth power is negative.	Not a real number.

ⓒ

Calculation of the fifth root of -243, including the expression, explanation, and final solution.
	$\sqrt[5]{−243}$
Since ${(−3)}^{5} = −243 .$	$−3$

When we worked with square roots that had variables in the radicand, we restricted the variables to non-negative values. Now we will remove this restriction.

The odd root of a number can be either positive or negative. We have seen that $\sqrt[3]{−64} = −4$ .

But the even root of a non-negative number is always non-negative, because we take the principal nth root.

Suppose we start with $a = −5$ .

\begin{matrix} {(−5)}^{4} = 625 & \sqrt[4]{625} = 5 \end{matrix}

How can we make sure the fourth root of −5 raised to the fourth power, ${(−5)}^{4}$ is 5? We will see in the following property.

Simplify: ⓐ $\sqrt{x^{2}}$ ⓑ $\sqrt[3]{n^{3}}$ ⓒ $\sqrt[4]{p^{4}}$ ⓓ $\sqrt[5]{y^{5}}$ .

Solution

We use the absolute value to be sure to get the positive root.

ⓐ

Illustration of simplifying sqrt(x^2) to |x| and the underlying mathematical justification.
	$\sqrt{x^{2}}$
Since ${(x)}^{2} = x^{2}$ and we want the positive root.	$\| x \|$

ⓑ

This table illustrates the simplification of the cube root of n cubed, providing the mathematical reasoning for the derivation.
	$\sqrt[3]{n^{3}}$
Since ${(n)}^{3} = n^{3} .$ It is an odd root so there is no need for an absolute value sign.	$n$

ⓒ

Illustrates the simplification of the fourth root of p to the fourth power to the absolute value of p, with an explanation.
	$\sqrt[4]{p^{4}}$
Since ${(p)}^{4} = p^{4}$ and we want the positive root.	$\| p \|$

ⓓ

This table demonstrates the simplification of the fifth root of y^5 to y, explaining why an absolute value is not required for odd roots.
	$\sqrt[5]{y^{5}}$
Since ${(y)}^{5} = y^{5} .$ It is an odd root so there is no need for an absolute value sign.	$y$

Simplify: ⓐ $\sqrt[3]{y^{18}}$ ⓑ $\sqrt[4]{z^{8}}$ .

Solution

ⓐ

This table demonstrates the step-by-step algebraic simplification of the cube root of y to the power of 18.
	$\sqrt[3]{y^{18}}$
Since ${(y^{6})}^{3} = y^{18} .$	$\sqrt[3]{{(y^{6})}^{3}}$
	$y^{6}$

ⓑ

Step-by-step simplification of the radical expression 4th root of z to the power of 8, showing mathematical reasons.
	$\sqrt[4]{z^{8}}$
Since ${(z^{2})}^{4} = z^{8} .$	$\sqrt[4]{{(z^{2})}^{4}}$
Since $z^{2}$ is positive, we do not need an absolute value sign.	$z^{2}$

Simplify: ⓐ $\sqrt[3]{64 p^{6}}$ ⓑ $\sqrt[4]{16 q^{12}}$ .

Solution

ⓐ

Illustrates the step-by-step process of simplifying a cube root expression with a perfect cube radicand.
	$\sqrt[3]{64 p^{6}}$
Rewrite $64 p^{6} as {(4 p^{2})}^{3} .$	$\sqrt[3]{{(4 p^{2})}^{3}}$
Take the cube root.	$4 p^{2}$

ⓑ

Steps to simplify the fourth root of 16q^12, showing intermediate algebraic expressions for each operation.
	$\sqrt[4]{16 q^{12}}$
Rewrite the radicand as a fourth power.	$\sqrt[4]{{(2 q^{3})}^{4}}$
Take the fourth root.	$2 \| q^{3} \|$

Use the Product Property to Simplify Expressions with Higher Roots

We will simplify expressions with higher roots in much the same way as we simplified expressions with square roots. An nth root is considered simplified if it has no factors of $m^{n}$ .

We will generalize the Product Property of Square Roots to include any integer root $n \geq 2$ .

Simplify: ⓐ $\sqrt[3]{x^{4}}$ ⓑ $\sqrt[4]{x^{7}}$ .

Solution

ⓐ

Steps for simplifying the cube root of x^4 by factoring the radicand and extracting perfect cubes.
	$\sqrt[3]{x^{4}}$
Rewrite the radicand as a product using the largest perfect cube factor.	$\sqrt[3]{x^{3} \cdot x}$
Rewrite the radical as the product of two radicals.	$\sqrt[3]{x^{3}} \cdot \sqrt[3]{x}$
Simplify.	$x \sqrt[3]{x}$

ⓑ

Illustrates the step-by-step simplification of the radical expression fourth root of x to the power of seven.
	$\sqrt[4]{x^{7}}$
Rewrite the radicand as a product using the greatest perfect fourth power factor.	$\sqrt[4]{x^{4} \cdot x^{3}}$
Rewrite the radical as the product of two radicals.	$\sqrt[4]{x^{4}} \cdot \sqrt[4]{x^{3}}$
Simplify.	$\| x \| \sqrt[4]{x^{3}}$

Simplify: ⓐ $\sqrt[3]{16}$ ⓑ $\sqrt[4]{243}$ .

Solution

ⓐ

Step-by-step process demonstrating the simplification of the cube root of 16.
	$\sqrt[3]{16}$
	$\sqrt[3]{2^{4}}$
Rewrite the radicand as a product using the greatest perfect cube factor.	$\sqrt[3]{2^{3} \cdot 2}$
Rewrite the radical as the product of two radicals.	$\sqrt[3]{2^{3}} \cdot \sqrt[3]{2}$
Simplify.	$2 \sqrt[3]{2}$

ⓑ

Step-by-step simplification of the fourth root of 243.
	$\sqrt[4]{243}$
	$\sqrt[4]{3^{5}}$
Rewrite the radicand as a product using the greatest perfect fourth power factor.	$\sqrt[4]{3^{4} \cdot 3}$
Rewrite the radical as the product of two radicals.	$\sqrt[4]{3^{4}} \cdot \sqrt[4]{3}$
Simplify.	$3 \sqrt[4]{3}$

Don’t forget to use the absolute value signs when taking an even root of an expression with a variable in the radical.

Simplify: ⓐ $\sqrt[3]{24 x^{7}}$ ⓑ $\sqrt[4]{80 y^{14}}$ .

Solution

ⓐ

This table details the step-by-step simplification of the cube root expression (24x^7)^(1/3) to its simplified form 2x^2 * (3x)^(1/3).
	$\sqrt[3]{24 x^{7}}$
Rewrite the radicand as a product using perfect cube factors.	$\sqrt[3]{2^{3} x^{6} \cdot 3 x}$
Rewrite the radical as the product of two radicals.	$\sqrt[3]{2^{3} x^{6}} \cdot \sqrt[3]{3 x}$
Rewrite the first radicand as ${(2 x^{2})}^{3} .$	$\sqrt[3]{{(2 x^{2})}^{3}} \cdot \sqrt[3]{3 x}$
Simplify.	$2 x^{2} \sqrt[3]{3 x}$

ⓑ

This table illustrates the step-by-step process of simplifying a fourth root radical expression.
	$\sqrt[4]{80 y^{14}}$
Rewrite the radicand as a product using perfect fourth power factors.	$\sqrt[4]{2^{4} y^{12} \cdot 5 y^{2}}$
Rewrite the radical as the product of two radicals.	$\sqrt[4]{2^{4} y^{12}} \cdot \sqrt[4]{5 y^{2}}$
Rewrite the first radicand as ${(2 y^{3})}^{4} .$	$\sqrt[4]{{(2 y^{3})}^{4}} \cdot \sqrt[4]{5 y^{2}}$
Simplify.	$2 \| y^{3} \| \sqrt[4]{5 y^{2}}$

Simplify: ⓐ $\sqrt[3]{−27}$ ⓑ $\sqrt[4]{−16}$ .

Solution

ⓐ

Steps to calculate the cube root of -27.
	$\sqrt[3]{−27}$
Rewrite the radicand as a product using perfect cube factors.	$\sqrt[3]{{(−3)}^{3}}$
Take the cube root.	$−3$

ⓑ

This table demonstrates why the fourth root of -16 is not considered a real number.
	$\sqrt[4]{−16}$
There is no real number $n$ where $n^{4} = −16 .$	Not a real number.

Use the Quotient Property to Simplify Expressions with Higher Roots

We can simplify higher roots with quotients in the same way we simplified square roots. First we simplify any fractions inside the radical.

Simplify: ⓐ $\sqrt[3]{\frac{a^{8}}{a^{5}}}$ ⓑ $\sqrt[4]{\frac{a^{10}}{a^{2}}}$ .

Solution

ⓐ

Step-by-step simplification of the radical expression ³√(a⁸/a⁵) to 'a', illustrating algebraic simplification.
	$\sqrt[3]{\frac{a^{8}}{a^{5}}}$
Simplify the fraction under the radical first.	$\sqrt[3]{a^{3}}$
Simplify.	$a$

ⓑ

Step-by-step process demonstrating the simplification of a radical expression involving variables.
	$\sqrt[4]{\frac{a^{10}}{a^{2}}}$
Simplify the fraction under the radical first.	$\sqrt[4]{a^{8}}$
Rewrite the radicand using perfect fourth power factors.	$\sqrt[4]{{(a^{2})}^{4}}$
Simplify.	$a^{2}$

Previously, we used the Quotient Property ‘in reverse’ to simplify square roots. Now we will generalize the formula to include higher roots.

Simplify: ⓐ $\frac{\sqrt[3]{−108}}{\sqrt[3]{2}}$ ⓑ $\frac{\sqrt[4]{96 x^{7}}}{\sqrt[4]{3 x^{2}}}$ .

Solution

ⓐ

Step-by-step simplification of a cube root expression using radical properties.
	$\frac{\sqrt[3]{−108}}{\sqrt[3]{2}}$
Neither radicand is a perfect cube, so use the Quotient Property to write as one radical.	$\sqrt[3]{\frac{−108}{2}}$
Simplify the fraction under the radical.	$\sqrt[3]{−54}$
Rewrite the radicand as a product using perfect cube factors.	$\sqrt[3]{{(−3)}^{3} \cdot 2}$
Rewrite the radical as the product of two radicals.	$\sqrt[3]{{(−3)}^{3}} \cdot \sqrt[3]{2}$
Simplify.	$−3 \sqrt[3]{2}$

ⓑ

Step-by-step simplification of a radical expression involving the division of fourth roots.
	$\frac{\sqrt[4]{96 x^{7}}}{\sqrt[4]{3 x^{2}}}$
Neither radicand is a perfect fourth power, so use the Quotient Property to write as one radical.	$\sqrt[4]{\frac{96 x^{7}}{3 x^{2}}}$
Simplify the fraction under the radical.	$\sqrt[4]{32 x^{5}}$
Rewrite the radicand as a product using perfect fourth power factors.	$\sqrt[4]{2^{4} x^{4} \cdot 2 x}$
Rewrite the radical as the product of two radicals.	$\sqrt[4]{{(2 x)}^{4}} \cdot \sqrt[4]{2 x}$
Simplify.	$2 \| x \| \sqrt[4]{2 x}$

If the fraction inside the radical cannot be simplified, we use the first form of the Quotient Property to rewrite the expression as the quotient of two radicals.

Simplify: ⓐ $\sqrt[3]{\frac{24 x^{7}}{y^{3}}}$ ⓑ $\sqrt[4]{\frac{48 x^{10}}{y^{8}}}$ .

Solution

ⓐ

Step-by-step process for simplifying a cube root expression involving a fraction and variables.
	$\sqrt[3]{\frac{24 x^{7}}{y^{3}}}$
The fraction in the radicand cannot be simplified. Use the Quotient Property to write as two radicals.	$\frac{\sqrt[3]{24 x^{7}}}{\sqrt[3]{y^{3}}}$
Rewrite each radicand as a product using perfect cube factors.	$\frac{\sqrt[3]{8 x^{6} \cdot 3 x}}{\sqrt[3]{y^{3}}}$
Rewrite the numerator as the product of two radicals.	$\frac{\sqrt[3]{{(2 x^{2})}^{3}} \sqrt[3]{3 x}}{\sqrt[3]{y^{3}}}$
Simplify.	$\frac{2 x^{2} \sqrt[3]{3 x}}{y}$

ⓑ

Step-by-step process demonstrating the simplification of a fourth root algebraic expression using radical properties.
	$\sqrt[4]{\frac{48 x^{10}}{y^{8}}}$
The fraction in the radicand cannot be simplified. Use the Quotient Property to write as two radicals.	$\frac{\sqrt[4]{48 x^{10}}}{\sqrt[4]{y^{8}}}$
Rewrite each radicand as a product using perfect fourth power factors.	$\frac{\sqrt[4]{16 x^{8} \cdot 3 x^{2}}}{\sqrt[4]{y^{8}}}$
Rewrite the numerator as the product of two radicals.	$\frac{\sqrt[4]{{(2 x^{2})}^{4}} \sqrt[4]{3 x^{2}}}{\sqrt[4]{{(y^{2})}^{4}}}$
Simplify.	$\frac{2 x^{2} \sqrt[4]{3 x^{2}}}{y^{2}}$

Add and Subtract Higher Roots

We can add and subtract higher roots like we added and subtracted square roots. First we provide a formal definition of like radicals.

Like radicals have the same index and the same radicand.

$9 \sqrt[4]{42 x}$ and $−2 \sqrt[4]{42 x}$ are like radicals.
$5 \sqrt[3]{125 x}$ and $6 \sqrt[3]{125 y}$ are not like radicals. The radicands are different.
$2 \sqrt[5]{1000 q}$ and $−4 \sqrt[4]{1000 q}$ are not like radicals. The indices are different.

We add and subtract like radicals in the same way we add and subtract like terms. We can add $9 \sqrt[4]{42 x} + (−2 \sqrt[4]{42 x})$ and the result is $7 \sqrt[4]{42 x}$ .

Simplify: ⓐ $\sqrt[3]{4 x} + \sqrt[3]{4 x}$ ⓑ $4 \sqrt[4]{8} - 2 \sqrt[4]{8}$ .

Solution

ⓐ

Example illustrating the addition of like radicals, including an explanation and the simplified expression.
	$\sqrt[3]{4 x} + \sqrt[3]{4 x}$
The radicals are like, so we add the coefficients.	$2 \sqrt[3]{4 x}$

ⓑ

Simplification of like radical expressions, demonstrating the subtraction of coefficients with an accompanying explanation.
	$4 \sqrt[4]{8} - 2 \sqrt[4]{8}$
The radicals are like, so we subtract the coefficients.	$2 \sqrt[4]{8}$

When an expression does not appear to have like radicals, we will simplify each radical first. Sometimes this leads to an expression with like radicals.

Simplify: ⓐ $\sqrt[3]{54} - \sqrt[3]{16}$ ⓑ $\sqrt[4]{48} + \sqrt[4]{243}$ .

Solution

ⓐ

Step-by-step simplification of an expression involving cube roots.
	$\sqrt[3]{54} - \sqrt[3]{16}$
Rewrite each radicand using perfect cube factors.	$\sqrt[3]{27} \cdot \sqrt[3]{2} - \sqrt[3]{8} \cdot \sqrt[3]{2}$
Rewrite the perfect cubes.	$\sqrt[3]{{(3)}^{3}} \sqrt[3]{2} - \sqrt[3]{{(2)}^{3}} \sqrt[3]{2}$
Simplify the radicals where possible.	$3 \sqrt[3]{2} - 2 \sqrt[3]{2}$
Combine like radicals.	$\sqrt[3]{2}$

ⓑ

Step-by-step simplification of a sum of fourth roots by factoring out perfect fourth powers and combining like radicals.
	$\sqrt[4]{48} + \sqrt[4]{243}$
Rewrite using perfect fourth power factors.	$\sqrt[4]{16} \cdot \sqrt[4]{3} + \sqrt[4]{81} \cdot \sqrt[4]{3}$
Rewrite the perfect fourth powers.	$\sqrt[4]{{(2)}^{4}} \sqrt[4]{3} + \sqrt[4]{{(3)}^{4}} \sqrt[4]{3}$
Simplify the radicals where possible.	$2 \sqrt[4]{3} + 3 \sqrt[4]{3}$
Combine like radicals.	$5 \sqrt[4]{3}$

Simplify: ⓐ $\sqrt[3]{24 x^{4}} - \sqrt[3]{−81 x^{7}}$ ⓑ $\sqrt[4]{162 y^{9}} + \sqrt[4]{512 y^{5}}$ .

Solution

ⓐ

Step-by-step simplification of an algebraic expression involving cube roots.
	$\sqrt[3]{24 x^{4}} - \sqrt[3]{−81 x^{7}}$
Rewrite each radicand using perfect cube factors.	$\sqrt[3]{8 x^{3}} \cdot \sqrt[3]{3 x} - \sqrt[3]{−27 x^{6}} \cdot \sqrt[3]{3 x}$
Rewrite the perfect cubes.	$\sqrt[3]{{(2 x)}^{3}} \sqrt[3]{3 x} - \sqrt[3]{{(−3 x^{2})}^{3}} \sqrt[3]{3 x}$
Simplify the radicals where possible.	$2 x \sqrt[3]{3 x} - (−3 x^{2} \sqrt[3]{3 x})$

ⓑ

Step-by-step simplification of a sum involving fourth-root radical expressions.
	$\sqrt[4]{162 y^{9}} + \sqrt[4]{512 y^{5}}$
Rewrite each radicand using perfect fourth power factors.	$\sqrt[4]{81 y^{8}} \cdot \sqrt[4]{2 y} + \sqrt[4]{256 y^{4}} \cdot \sqrt[4]{2 y}$
Rewrite the perfect fourth powers.	$\sqrt[4]{{(3 y^{2})}^{4}} \cdot \sqrt[4]{2 y} + \sqrt[4]{{(4 y)}^{4}} \cdot \sqrt[4]{2 y}$
Simplify the radicals where possible.	$3 y^{2} \sqrt[4]{2 y} + 4 \| y \| \sqrt[4]{2 y}$

Key Concepts

Properties of
an when n is an even number and
- $a \geq 0$ , then $\sqrt[n]{a}$ is a real number
- $a < 0$ , then $\sqrt[n]{a}$ is not a real number
- When $n$ is an odd number, $\sqrt[n]{a}$ is a real number for all values of a.
- For any integer $n \geq 2$ , when n is odd $\sqrt[n]{a^{n}} = a$
- For any integer $n \geq 2$ , when n is even $\sqrt[n]{a^{n}} = | a |$
$\sqrt[n]{a}$ is considered simplified if a has no factors of $m^{n}$ .
Product Property of nth Roots

$\sqrt[n]{a b} = \sqrt[n]{a} \cdot \sqrt[n]{b} and \sqrt[n]{a} \cdot \sqrt[n]{b} = \sqrt[n]{a b}$
Quotient Property of nth Roots

$\sqrt[n]{\frac{a}{b}} = \frac{\sqrt[n]{a}}{\sqrt[n]{b}} and \frac{\sqrt[n]{a}}{\sqrt[n]{b}} = \sqrt[n]{\frac{a}{b}}$
To combine like radicals, simply add or subtract the coefficients while keeping the radical the same.

Practice Makes Perfect

Simplify Expressions with Higher Roots

In the following exercises, simplify.

ⓐ $\sqrt[3]{216}$ ⓑ $\sqrt[4]{256}$ ⓒ $\sqrt[5]{32}$

ⓐ $\sqrt[3]{27}$ ⓑ $\sqrt[4]{16}$ ⓒ $\sqrt[5]{243}$

Solution

ⓐ $3$ ⓑ $2$ ⓒ $3$

ⓐ $\sqrt[3]{512}$ ⓑ $\sqrt[4]{81}$ ⓒ $\sqrt[5]{1}$

ⓐ $\sqrt[3]{125}$ ⓑ $\sqrt[4]{1296}$ ⓒ $\sqrt[5]{1024}$

Solution

ⓐ $5$ ⓑ $6$ ⓒ $4$

ⓐ $\sqrt[3]{−8}$ ⓑ $\sqrt[4]{−81}$ ⓒ $\sqrt[5]{−32}$

ⓐ $\sqrt[3]{−64}$ ⓑ $\sqrt[4]{−16}$ ⓒ $\sqrt[5]{−243}$

Solution

ⓐ $−4$ ⓑ $not real$ ⓒ $−3$

ⓐ $\sqrt[3]{−125}$ ⓑ $\sqrt[4]{−1296}$ ⓒ $\sqrt[5]{−1024}$

ⓐ $\sqrt[3]{−512}$ ⓑ $\sqrt[4]{−81}$ ⓒ $\sqrt[5]{−1}$

Solution

ⓐ $−8$ ⓑ not a real number ⓒ $−1$

ⓐ $\sqrt[5]{u^{5}}$ ⓑ $\sqrt[8]{v^{8}}$

ⓐ $\sqrt[3]{a^{3}}$
ⓑ

Solution

ⓐ $a$ ⓑ $| b |$

ⓐ $\sqrt[4]{y^{4}}$ ⓑ $\sqrt[7]{m^{7}}$

ⓐ $\sqrt[8]{k^{8}}$ ⓑ $\sqrt[6]{p^{6}}$

Solution

ⓐ $| k |$ ⓑ $| p |$

ⓐ $\sqrt[3]{x^{9}}$ ⓑ $\sqrt[4]{y^{12}}$

ⓐ $\sqrt[5]{a^{10}}$ ⓑ $\sqrt[3]{b^{27}}$

Solution

ⓐ $a^{2}$ ⓑ $b^{9}$

ⓐ $\sqrt[4]{m^{8}}$ ⓑ $\sqrt[5]{n^{20}}$

ⓐ $\sqrt[6]{r^{12}}$ ⓑ $\sqrt[3]{s^{30}}$

Solution

ⓐ $r^{2}$ ⓑ $s^{10}$

ⓐ $\sqrt[4]{16 x^{8}}$ ⓑ $\sqrt[6]{64 y^{12}}$

ⓐ $\sqrt[3]{−8 c^{9}}$ ⓑ $\sqrt[3]{125 d^{15}}$

Solution

ⓐ $−2 c^{3}$ ⓑ $5 d^{5}$

ⓐ $\sqrt[3]{216 a^{6}}$ ⓑ $\sqrt[5]{32 b^{20}}$

ⓐ $\sqrt[7]{128 r^{14}}$ ⓑ $\sqrt[4]{81 s^{24}}$

Solution

ⓐ $2 r^{2}$ ⓑ $3 s^{6}$

Use the Product Property to Simplify Expressions with Higher Roots

In the following exercises, simplify.

ⓐ $\sqrt[3]{r^{5}}$ ⓑ $\sqrt[4]{s^{10}}$

ⓐ $\sqrt[5]{u^{7}}$ ⓑ $\sqrt[6]{v^{11}}$

Solution

ⓐ $u \sqrt[5]{u^{2}}$ ⓑ $|v| \sqrt[6]{v^{5}}$

ⓐ $\sqrt[4]{m^{5}}$ ⓑ $\sqrt[8]{n^{10}}$

ⓐ $\sqrt[5]{p^{8}}$ ⓑ $\sqrt[3]{q^{8}}$

Solution

ⓐ $p \sqrt[5]{p^{3}}$ ⓑ $q^{2} \sqrt[3]{q^{2}}$

ⓐ $\sqrt[4]{32}$ ⓑ $\sqrt[7]{64}$

ⓐ $\sqrt[3]{625}$ ⓑ $\sqrt[6]{128}$

Solution

ⓐ $5 \sqrt[3]{5}$ ⓑ $2 \sqrt[6]{2}$

ⓐ $\sqrt[5]{64}$ ⓑ $\sqrt[3]{256}$

ⓐ $\sqrt[4]{3125}$ ⓑ $\sqrt[3]{81}$

Solution

ⓐ $5 \sqrt[4]{5}$ ⓑ $3 \sqrt[3]{3}$

ⓐ $\sqrt[3]{108 x^{5}}$ ⓑ $\sqrt[4]{48 y^{6}}$

ⓐ $\sqrt[5]{96 a^{7}}$ ⓑ $\sqrt[3]{375 b^{4}}$

Solution

ⓐ $2 a \sqrt[5]{3 a^{2}}$ ⓑ $5 b \sqrt[3]{3 b}$

ⓐ $\sqrt[4]{405 m^{10}}$ ⓑ $\sqrt[5]{160 n^{8}}$

ⓐ $\sqrt[3]{512 p^{5}}$ ⓑ $\sqrt[4]{324 q^{7}}$

Solution

ⓐ $8 p \sqrt[3]{p^{2}}$ ⓑ $3 |q| \sqrt[4]{4 q^{3}}$

ⓐ $\sqrt[3]{−864}$ ⓑ $\sqrt[4]{−256}$

ⓐ $\sqrt[5]{−486}$ ⓑ $\sqrt[6]{−64}$

Solution

ⓐ $−3 \sqrt[5]{2}$ ⓑ $not real$

ⓐ $\sqrt[5]{−32}$ ⓑ $\sqrt[8]{−1}$

ⓐ $\sqrt[3]{−8}$ ⓑ $\sqrt[4]{−16}$

Solution

ⓐ $−2$ ⓑ $not real$

Use the Quotient Property to Simplify Expressions with Higher Roots

In the following exercises, simplify.

ⓐ $\sqrt[3]{\frac{p^{11}}{p^{2}}}$ ⓑ $\sqrt[4]{\frac{q^{17}}{q^{13}}}$

ⓐ $\sqrt[5]{\frac{d^{12}}{d^{7}}}$ ⓑ $\sqrt[8]{\frac{m^{12}}{m^{4}}}$

Solution

ⓐ $d$ ⓑ $| m |$

ⓐ $\sqrt[5]{\frac{u^{21}}{u^{11}}}$ ⓑ $\sqrt[6]{\frac{v^{30}}{v^{12}}}$

ⓐ $\sqrt[3]{\frac{r^{14}}{r^{5}}}$ ⓑ $\sqrt[4]{\frac{c^{21}}{c^{9}}}$

Solution

ⓐ $r^{3}$ ⓑ $| c^{3} |$

ⓐ $\frac{\sqrt[4]{64}}{\sqrt[4]{2}}$ ⓑ $\frac{\sqrt[5]{128 x^{8}}}{\sqrt[5]{2 x^{2}}}$

ⓐ $\frac{\sqrt[3]{−625}}{\sqrt[3]{5}}$ ⓑ $\frac{\sqrt[4]{80 m^{7}}}{\sqrt[4]{5 m}}$

Solution

ⓐ $−5$ ⓑ $2 |m| \sqrt[4]{m^{2}}$

ⓐ $\sqrt[3]{\frac{1250}{2}}$ ⓑ $\sqrt[4]{\frac{486 y^{9}}{2 y^{3}}}$

ⓐ $\sqrt[3]{\frac{162}{6}}$ ⓑ $\sqrt[4]{\frac{160 r^{10}}{5 r^{3}}}$

Solution

ⓐ $3 \sqrt[3]{6}$ ⓑ $2 | r | \sqrt[4]{2 r^{3}}$

ⓐ $\sqrt[3]{\frac{54 a^{8}}{b^{3}}}$ ⓑ $\sqrt[4]{\frac{64 c^{5}}{d^{2}}}$

ⓐ $\sqrt[5]{\frac{96 r^{11}}{s^{3}}}$ ⓑ $\sqrt[6]{\frac{128 u^{7}}{v^{3}}}$

Solution

ⓐ $2 r^{2} \sqrt[5]{\frac{3 r}{s^{3}}}$ ⓑ $2 |u| \sqrt[6]{\frac{2 u}{v^{3}}}$

ⓐ $\sqrt[3]{\frac{81 s^{8}}{t^{3}}}$ ⓑ $\sqrt[4]{\frac{64 p^{15}}{q^{12}}}$

ⓐ $\sqrt[3]{\frac{625 u^{10}}{v^{3}}}$ ⓑ $\sqrt[4]{\frac{729 c^{21}}{d^{8}}}$

Solution

ⓐ $\frac{5 u^{3} \sqrt[3]{5 u}}{v}$ ⓑ $\frac{3 |c^{5}| \sqrt[4]{9 c}}{d^{2}}$

Add and Subtract Higher Roots

In the following exercises, simplify.

ⓐ $\sqrt[7]{8 p} + \sqrt[7]{8 p}$ ⓑ $3 \sqrt[3]{25} - \sqrt[3]{25}$

ⓐ $\sqrt[3]{15 q} + \sqrt[3]{15 q}$ ⓑ $2 \sqrt[4]{27} - 6 \sqrt[4]{27}$

Solution

ⓐ $2 \sqrt[3]{15 q}$ ⓑ $−4 \sqrt[4]{27}$

ⓐ $3 \sqrt[5]{9 x} + 7 \sqrt[5]{9 x}$ ⓑ $8 \sqrt[7]{3 q} - 2 \sqrt[7]{3 q}$

ⓐ An algebraic expression showing the sum of 23 times the 12th root of 4y and 19 times the 12th root of 4y. The expression is 23(12th root of 4y) + 19(12th root of 4y). ⓑ The mathematical expression 31 times the 10th root of 5z minus 17 times the 10th root of 5z is displayed.

Solution

ⓐ The image displays the mathematical expression 42 times the 12th root of 4y. ⓑ The image shows the mathematical expression 14 times the 10th root of 5z.

ⓐ $\sqrt[3]{81} - \sqrt[3]{192}$ ⓑ $\sqrt[4]{512} - \sqrt[4]{32}$

ⓐ $\sqrt[3]{250} - \sqrt[3]{54}$ ⓑ $\sqrt[4]{243} - \sqrt[4]{1875}$

Solution

ⓐ $2 \sqrt[3]{2}$ ⓑ $−2 \sqrt[4]{3}$

ⓐ $\sqrt[3]{128} + \sqrt[3]{250}$ ⓑ $\sqrt[5]{729} + \sqrt[5]{96}$

ⓐ $\sqrt[4]{243} + \sqrt[4]{1250}$ ⓑ $\sqrt[3]{2000} + \sqrt[3]{54}$

Solution

ⓐ $3 \sqrt[4]{3} + 5 \sqrt[4]{2}$ ⓑ $13 \sqrt[3]{2}$

ⓐ $\sqrt[3]{64 a^{10}} - \sqrt[3]{−216 a^{12}}$ ⓑ $\sqrt[4]{486 u^{7}} + \sqrt[4]{768 u^{3}}$

ⓐ $\sqrt[3]{80 b^{5}} - \sqrt[3]{−270 b^{3}}$ ⓑ $\sqrt[4]{160 v^{10}} - \sqrt[4]{1280 v^{3}}$

Solution

ⓐ $2 b \sqrt[3]{10 b^{2}} + 3 b \sqrt[3]{10}$ ⓑ $2 v^{2} \sqrt[4]{10 v^{2}} - 4 \sqrt[4]{5 v^{3}}$

Mixed Practice

In the following exercises, simplify.

$\sqrt[4]{16}$

$\sqrt[6]{64}$

Solution

$2$

$\sqrt[3]{a^{3}}$

A mathematical expression featuring the 12th root of b raised to the power of 12. This expression simplifies to 'b'.

Solution

$| b |$

$\sqrt[3]{−8 c^{9}}$

$\sqrt[3]{125 d^{15}}$

Solution

$5 d^{5}$

$\sqrt[3]{r^{5}}$

$\sqrt[4]{s^{10}}$

Solution

$s^{2} \sqrt[4]{s^{2}}$

$\sqrt[3]{108 x^{5}}$

$\sqrt[4]{48 y^{6}}$

Solution

$2 |y| \sqrt[4]{3 y^{2}}$

$\sqrt[5]{−486}$

$\sqrt[6]{−64}$

Solution

$not real$

$\frac{\sqrt[4]{64}}{\sqrt[4]{2}}$

$\frac{\sqrt[5]{128 x^{8}}}{\sqrt[5]{2 x^{2}}}$

Solution

$2 x \sqrt[5]{2 x}$

$\sqrt[5]{\frac{96 r^{11}}{s^{3}}}$

$\sqrt[6]{\frac{128 u^{7}}{v^{3}}}$

Solution

$2 |u| \sqrt[6]{\frac{2 u}{v^{3}}}$

$\sqrt[3]{81} - \sqrt[3]{192}$

$\sqrt[4]{512} - \sqrt[4]{32}$

Solution

$2 \sqrt[4]{2}$

$\sqrt[3]{64 a^{10}} - \sqrt[3]{−216 a^{12}}$

$\sqrt[4]{486 u^{7}} + \sqrt[4]{768 u^{3}}$

Solution

$3 │ u │ \sqrt[4]{6 u^{3}} + 4 \sqrt[4]{3 u^{3}}$

Everyday Math

Population growth The expression $10 \cdot x^{n}$ models the growth of a mold population after $n$ generations. There were 10 spores at the start, and each had $x$ offspring. So $10 \cdot x^{5}$ is the number of offspring at the fifth generation. At the fifth generation there were 10,240 offspring. Simplify the expression $\sqrt[5]{\frac{10,240}{10}}$ to determine the number of offspring of each spore.

Spread of a virus The expression $3 \cdot x^{n}$ models the spread of a virus after $n$ cycles. There were three people originally infected with the virus, and each of them infected $x$ people. So $3 \cdot x^{4}$ is the number of people infected on the fourth cycle. At the fourth cycle 1875 people were infected. Simplify the expression $\sqrt[4]{\frac{1875}{3}}$ to determine the number of people each person infected.

Solution

$5$

Writing Exercises

Explain how you know that $\sqrt[5]{x^{10}} = x^{2}$ .

Explain why $\sqrt[4]{−64}$ is not a real number but $\sqrt[3]{−64}$ is.

Solution

Answers may vary.

Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

This table has four columns and five rows. The first row labels each column: “I can…,” “Confidentaly,” “With some help,” and “No – I don’t get it!” The rows under the “I can…,” column read, “simplify expressions with hither roots.,” “use the product property to simplify expressions with higher roots.,” “use the quotient property to simplify expressions with higher roots.,” and “add and subtract higher roots.” The rest of the rows under the columns are empty.

ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve?

nth root of a number

b^{n} = a

, then

b

is an nth root of

a

principal nth root

The principal nth root of

a

is written

\sqrt[n]{a}

index

\sqrt[n]{a}

n is called the index of the radical.

like radicals

Radicals with the same index and same radicand are called like radicals.